Rectangular multiple-relaxation-time lattice Boltzmann method for the Navier-Stokes and nonlinear...

Area of Science:

• Computational fluid dynamics
• Numerical methods for partial differential equations
• Mesoscopic physics

Background:

• The multiple-relaxation-time lattice Boltzmann (MRT-LB) method is a powerful tool for simulating fluid dynamics and diffusion.
• Existing MRT-LB methods often rely on specific lattice structures, limiting their applicability.
• Anisotropy in rectangular lattices can lead to inconsistencies with physical models like Navier-Stokes equations.

Purpose of the Study:

• To develop a general rectangular multiple-relaxation-time lattice Boltzmann (RMRT-LB) method.
• To extend the unified framework of MRT-LB to handle Navier-Stokes equations (NSEs) and nonlinear convection-diffusion equations (NCDEs) on rectangular lattices.
• To address and resolve the inconsistency arising from lattice anisotropy in NSEs.

Main Methods:

• Extension of the unified MRT-LB framework to utilize an equilibrium distribution function (EDF) on a rectangular lattice.
• Modification of the relaxation matrix to account for dynamic and bulk viscosities, resolving anisotropy issues.
• Direct Taylor expansion method for macroscopic equation recovery and direct application of the unified framework for NCDEs.

Main Results:

• The RMRT-LB method successfully recovers the macroscopic Navier-Stokes equations.
• The unified framework is directly applicable to the nonlinear convection-diffusion equation without modification due to lattice structure.
• Numerical simulations demonstrate accurate results and good numerical stability for the proposed RMRT-LB method.

Conclusions:

• The developed RMRT-LB method provides a versatile approach for simulating NSEs and NCDEs on rectangular lattices.
• The method offers accurate predictions and robust numerical stability, outperforming previous models.
• The RMRT-LB method generalizes existing MRT-LB models, with standard DdQq lattice models being special cases.